We describe a strongly minimal theory S in an effective language such that, in the chain of countable models of S, only the second model has a computable presentation. Thus there is a spectrum of an $ \omega_{1}^{}$-categorical theory which is neither upward nor downward closed. We also give an upper bound on the complexity of spectra.

We develop an account of logical consequence for the second-order language of set theory in the spirit of Boolos's plural interpretation of monadic second-order logic.

Kripke completeness of some infinitary predicate modal logics is presented. More precisely, we prove that if a normal modal logic $ \bf L$ above $ \bf K$ is $ \cal {D}$-persistent and universal, the infinitary and predicate extension of $ \bf L$ with BF_{$\scriptstyle \omega_{{1}}$} and BF is Kripke complete, where BF_{$\scriptstyle \omega_{{1}}$} and BF denote the formulas $ \bigwedge_{{{i\in\omega}}}^{{}}$ $ \Box$ p_i $ \supset$ $ \Box$ $ \bigwedge_{{{i\in\omega}}}^{{}}$p_i and $ \forall x \Box \varphi \supset \Box \forall x \varphi$, respectively. The results include the completeness of extensions of standard modal logics such as $ \bf K$, and its extensions by the schemata T, B, 4, 5, D, and their combinations. The proof is obtained by extending the correspondence between the representation of modal algebras and the completeness of propositional modal logic to infinite.

Using work of Devlin and Schindler in conjunction with work on Prikry forcing in a choiceless context done by the author, we show that two choiceless cardinal patterns have consistency strength of at least one Woodin cardinal.

It is shown that in ZF Martin's $ \aleph_{0}^{}$-axiom together with the axiom of countable choice for finite sets imply that arbitrary powers 2^X of a 2-point discrete space are Baire; and that the latter property implies the following: (a) the axiom of countable choice for finite sets, (b) power sets of infinite sets are Dedekind-infinite, (c) there are no amorphous sets, and (d) weak forms of the Kinna-Wagner principle.

In this paper we consider the theory of predicate logics in which the principle of bivalence or the principle of noncontradiction or both fail. Such logics are partial or paraconsistent or both. We consider sequent calculi for these logics and prove model existence. For $ \bf L_4$, the most general logic under consideration, we also prove a version of the Craig-Lyndon Interpolation Theorem. The paper shows that many techniques used for classical predicate logic generalize to partial and paraconsistent logics once the right setup is chosen. Our logic $ \bf L_4$ has a semantics that also underlies Belnap's logic and is related to the logic of bilattices. $ \bf L_4$ is in focus most of the time, but it is also shown how results obtained for $ \bf L_4$ can be transferred to several variants.

This paper discusses how to define logics as deductive limits of sequences of other logics. The case of da Costa's hierarchy of increasingly weaker paraconsistent calculi, known as $ \mathcal {C}$_n, 1 $ \leq$ n $ \leq$ $ \omega$, is carefully studied. The calculus $ \mathcal {C}$_{$\scriptstyle \omega$}, in particular, constitutes no more than a lower deductive bound to this hierarchy and differs considerably from its companions. A long standing problem in the literature (open for more than 35 years) is to define the deductive limit to this hierarchy, that is, its greatest lower deductive bound. The calculus $ \mathcal {C}$_min, stronger than $ \mathcal {C}$_{$\scriptstyle \omega$}, is first presented as a step toward this limit. As an alternative to the bivaluation semantics of $ \mathcal {C}$_min presented thereupon, possible-translations semantics are then introduced and suggested as the standard technique both to give this calculus a more reasonable semantics and to derive some interesting properties about it. Possible-translations semantics are then used to provide both a semantics and a decision procedure for $\mathcal{C}_{\rm Lim}$, the real deductive limit of da Costa's hierarchy. Possible-translations semantics also make it possible to characterize a precise sense of duality: as an example, $ \mathcal {D}$_min is proposed as the dual to $ \mathcal {C}$_min.

An account of Aristotle's syllogistic (including a full square of opposition and allowing for empty nouns) as an integral part of first-order predicate logic is lacking. Some say it is not possible. It is not found in the tradition stemming from Łukasiewicz's attempt nor in less formal approaches such as Strawson's. The Łukasiewicz tradition leaves Aristotle's syllogistic as an autonomous axiomatized system. In this paper Aristotle's syllogistic is presented within first-order predicate logic with special restricted quantifiers. The theory is not motivated primarily by historical considerations but as an accurate account of categorical sentences along lines suggested by recent work on natural language quantifiers and themes from supposition theory. It provides logical forms which conform to grammatical ones and is intended as a rival to accounts of quantifiers in natural language that appeal to binary quantifiers, for example, Wiggins or to restricted quantifiers, for example, Neale.

Parry discusses an extension of Aristotle's syllogistic that uses four nontraditional quantifiers. We show that his conjectured decision procedure for validity for the extended syllogistic is correct even if syllogisms have more than two premises. And we axiomatize this extension of the syllogistic.

After briefly considering the ancient Greek and nineteenth-century history of incommensurables (magnitudes that do not have a common aliquot part) and incomparables (magnitudes such that the larger can never be surpassed by any finite number of additions of the smaller to itself), this paper undertakes two tasks. The first task is to consider whether the numerical accommodation of incommensurables by means of the extension of the ordered field of rational numbers to the field of reals is `similar' or analogous to the numerical accommodation of incomparables by means of the extension of the ordered field of reals to the field of hyperreals. The second task is to evaluate several contemporary attempts to use concepts and techniques of the nonstandard mathematics of hyperreals to address classical, Zenonian puzzles concerning continuous magnitudes. The result of both these undertakings is, in a certain sense, `deflationary'.